Life expectations are essential inputs for many important personal decisions. We propose that longevity beliefs are responses constructed at the time of judgment, subject to irrelevant task and context factors, and leading to predictable biases. Specifically, we examine whether life expectancy is affected by the framing of expectations questions as either live-to or die-by, as well as by factors that actually affect longevity such as age, gender, and self-reported health. We find that individuals in a live-to frame report significantly higher chances of being alive at ages 55 through 95 than people in a corresponding die-by frame. Estimated mean life expectancies across three studies and 2300 respondents were 7.38 to 9.17 years longer when solicited in a live-to frame. We are additionally able to show how this framing works on a process level and how it affects preference for life annuities. Implications for models of financial decision making are discussed.
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Support for this research was provided by Alfred P. Sloan and Russell Sage Foundations. Additional support was provided by the Fuqua School of Business, Duke University and NIA Grant 5R01AG027934 to Columbia University.
Description of Weibull estimation procedure, including ability to calculate estimated value of annuity per respondent
Given that we had collected a series of survival probabilities for each participant at a sequence of specific ages, we wished to use this data to calculate a mean life expectancy per individual. A commonly used distribution in survival analysis is the Weibull function, which has desirable properties of being relatively flat in early years but then yields an increasing failure rate with increasing age, consistent with human aging patterns. The cumulative distribution function of the Weibull, which can be interpreted as probability of failure (death) at age x, is given as:
However, since our data are coded as probability of living at any age x, we use 1 – F(x) in our estimates and the equations below.
We start with the series of probability estimates p ij for participant i where x j represents the age for which the estimate was collected. In all of our studies, we collect probability estimates for ages 65, 75, 85, and 95; for some studies, we also collect estimates for age 55. We also have the participant’s current age, age i . We then assume that these probability estimates are consistent with the Weibull function in the following manner:
where i and βi are the Weibull distribution parameters specific to that particular individual.
The p ij values per participant, along with the current age, provide us with enough data points to estimate i and βi per individual. Furthermore, to “cap” the survival function at a reasonable upper bound, we assume p ij = 0% at age x j = 130 years. We selected age 130 as the upper bound based on the longest documented human lifespan: Jeanne Calment of France, who died at more than 122 years old. Given that many participants in our studies have grown up in a time of significant health and medical advances, it is not unreasonable to expect that some of them may make it to 130.
To estimate the Weibull parameters, we rearrange terms and perform log transforms to get the following linear expression:
We can now do a regression with the left hand side expression as Y and ln(x j –age i ) as X, which provides us with a constant and a coefficient per individual. Putting these estimates back into the expression above allows us to solve for αi and βi. Finally, to get mean expected age per individual, we set p ij = 50% and solve for x, which gives:
We can also use the Weibull estimates to generate the expected value of an annuity for each individual. To do so, we need to calculate the area under the curve for the Weibull function, per individual, for the years from age 65 until death (infinity). This integral comes out as:
where Γ is the Gamma function. The result of this calculation can then be multiplied by the per period value of the annuity; for our analysis, we use an annual value of $6,800 to be consistent with our survey stimuli.
As an example of this procedure, consider the data provided from one of our participants in Study 2. This individual provided estimates of 90% chance of being alive at age 65, 50% chance at age 75, 25% chance at 85, and 5% chance at age 95. To this list, we include a 0% chance at age 130. This participant’s current age was 47. Using the procedure outlined above, we estimate αi = 34 and βi = 3.4. The graph below shows the plot for the resultant function along with the original estimates from the participant. Using this function, the mean expected life expectancy for this individual is 77.6 years, and the expected value of the annuity that starts at age 65 is $87,485.
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Payne, J.W., Sagara, N., Shu, S.B. et al. Life expectancy as a constructed belief: Evidence of a live-to or die-by framing effect. J Risk Uncertain 46, 27–50 (2013). https://doi.org/10.1007/s11166-012-9158-0
- Life Expectancy
- Framing Effects
- D03 – Behavioral Economics
- D84 – Expectations